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Explicit formulae for stationary distributions of stress release processes

Part of: Markov processes Other natural sciences

Published online by Cambridge University Press:  14 July 2016

K. Borovkov  and
D. Vere-Jones
K. Borovkov*
Affiliation:
University of Melbourne
D. Vere-Jones*
Affiliation:
Victoria University of Wellington
*
∗Postal address: Department of Mathematics and Statistics, University of Melbourne, Parkville 3052, Australia. Email address: k.borovkov@ms.unimelb.edu.au
∗∗Postal address: School of Mathematical and Computer Sciences, Victoria University, PO Box 600, Wellington, New Zealand
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Abstract

Stress release processes are special Markov models attempting to describe the behaviour of stress and occurrence of earthquakes in seismic zones. The stress is built up linearly by tectonic forces and released spontaneously when earthquakes occur. Assuming that the risk is an exponential function of the stress, we derive closed form expressions for the stationary distribution of such processes, the moments of the risk, and the autocovariance function of the reciprocal risk process.

Keywords

Stress release modelsMarkov processesstationary distribution

MSC classification

Primary: 60J25: Continuous-time Markov processes on general state spaces
Secondary: 92F05: Other natural sciences (should also be assigned at least one other classification number in section 92)
Type
Research Papers
Information
Journal of Applied Probability , Volume 37 , Issue 2 , June 2000 , pp. 315 - 321
Copyright
Copyright © by the Applied Probability Trust 2000 

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References

Borovkov, A. A. (1998). Probability Theory. Gordon and Breach, New York.Google Scholar
Davis, M. H. A. (1984). Piecewise deterministic Markov processes: a general class of non-diffusion stochastic models. J. Roy. Statist. Soc. B 46, 353–388.Google Scholar
Erdelyi, A. (1956). Asymptotic Expansions. Dover, New York.Google Scholar
Gutenberg, B., and Richter, C. F. (1954). Seismicity of the Earth and Associated Phenomena, 2nd edn. Princeton University Press, Princeton, NJ.Google Scholar
Kagan, Y. Y. (1997). Seismic-moment frequency relations for shallow earthquakes: regional comparison. J. Geophys. Res. 102, 2835–2852.Google Scholar
Ogata, Y., and Vere-Jones, D. (1984). Inference for earthquake models: a self-correcting model. Stoch. Proc. Appl. 17, 337–347.Google Scholar
Rao, C. R., and Shanbhag, D. N. (1994). Choquet–Deny Type Functional Equations with Applications to Stochastic Models. Wiley, New York.Google Scholar
Vere-Jones, D. (1988). On the variance properties of stress release models. Austral. J. Statist. 30A, 123–135.Google Scholar
Vere-Jones, D., and Deng, Y. L. (1988). A point process analysis of historical earthquakes from North China. Engl. transl.: Earthquake Research in China 2, 165–181.Google Scholar
Vere-Jones, D., and Ogata, Y. (1984). On the moments of self-correcting process. J. Appl. Prob. 21, 335–342.CrossRefGoogle Scholar
Widder, D. V. (1941). The Laplace Transform. Princeton University Press, Princeton, NJ.Google Scholar
Zheng, X. (1991). Ergodic theorems for stress release processes. Stoch. Proc. Appl. 37, 239–258.Google Scholar
Zheng, X., and Vere-Jones, D. (1991). Applications of stress release models to historical earthquakes from North China. Pure Appl. Geophys. 135, 559–576.Google Scholar
Zheng, X., and Vere-Jones, D. (1994). Further applications of the stress release model to historical earthquake data. Tectonophysics 229, 101–121.Google Scholar